My Concorde is an LX model with the 3

So are you saying it cranks too slowly to start unless you use jumper cables? If that is correct, start by measuring the battery voltage. It should be 12.6 volts if it is fully charged.

It appears you have two different problems that are likely not related. First of all, I overlooked the obvious. Since the transmission was removed, it is almost certain the battery had to be disconnected. At that point the Engine Computer lost its memory, most importantly, "minimum throttle". Until it relearns that, idle speed will be too low for reliable starting, you won't get the normal "idle flare-up" to 1500 rpm at engine start-up, and it will typically stall when coming to a stop.

To get the engine to start you may need to hold the gas pedal down about 1/4". To meet the conditions for the relearn to take place, drive at highway speed with the engine warmed up, then coast for at least seven seconds without touching the pedals.

If the intermittent clicking you're hearing is a rather loud single click each time you turn the ignition switch to "crank", the first thing I'd suspect is a loose cable connection on the starter. The mechanic may have removed it to get it out of the way for removing the transmission, and forgot to tighten it.

"and still nothing"

What is "nothing"? Does it just make a single rather loud clunk each time the ignition switch is turned to "crank"? Does it crank but not start? If it is cranking normally, check for diagnostic fault codes first. Cycle the ignition switch three times from "off" to "run" without cranking the engine, then watch the code numbers show up in the odometer display.

The first thing to check is fuel pressure. That is one thing that is not monitored by the Engine Computer. GM fuel pumps most commonly fail while you're driving leaving you stranded on the side of he highway. Chrysler pumps almost never quit once they're running. When they fail they fail to start up. If you don't hear it run for one second after turning on the ignition switch, bang on the bottom of the tank a few times, possibly while a helper is cranking the engine.

The next thing is to connect a scanner to view live data. Under the "No start" menu it will show all the relevant switches, relays, and sensors and what the Engine Computer is seeing. In particular the MAP sensor should show the correct barometric pressure and a vacuum reading during cranking. The ASD relay request should be "On", and the camshaft position sensor and crankshaft position sensor should both show "present" or "in sync".

Unlock either front door with the key.

If the alarm won't reset from either front door, I would suspect broken wires between the door hinges or a switch has become disconnected from the key cylinder. I have run into defective switches on other car models many years ago, but broken wires are a lot more common.